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Abstract

We study, analytically and numerically, the dynamics of interband transitions in two-dimensional hexagonal periodic photonic lattices. We develop an analytical approach employing the Bragg resonances of different types and derive the effective multi-level models of the Landau-Zener-Majorana type. For two-dimensional periodic potentials without a tilt, we demonstrate the possibility of the Rabi oscillations between the resonant Fourier amplitudes. In a biased lattice, i.e., for a two-dimensional periodic potential with an additional linear tilt, we identify three basic types of the interband transitions or Zener tunnelling. First, this is a quasi-one-dimensional tunnelling that involves only two Bloch bands and occurs when the Bloch index crosses the Bragg planes away from one of the high-symmetry points. In contrast, at the high-symmetry points (i.e., at the M and Γ points), the Zener tunnelling is essentially two-dimensional, and it involves either three or six Bloch bands being described by the corresponding multi-level Landau-Zener-Majorana systems. We verify our analytical results by numerical simulations and observe an excellent agreement. Finally, we show that phase dislocations, or optical vortices, can tunnel between the spectral bands preserving their topological charge. Our theory describes the propagation of light beams in fabricated or optically-induced two-dimensional photonic lattices, but it can also be applied to the physics of cold atoms and Bose-Einstein condensates tunnelling in tilted two-dimensional optical potentials and other types of resonant wave propagation in periodic media.

Figures (8)

Fig. 1. (a) The first Brillouin zone of the hexagonal lattice with b=|bl|=2, where the vectors bl correspond to translations between the points M, M′ and M″. (b) The hexagonal cos-lattice Eq. (5) and (c) corresponding “triangular” sin-lattice. The vectors d1 and d2 give two fundamental periods of the lattice. (d) The Bloch band structure (the first 9 bands) of the hexagonal cos-lattice in (b) with I0=0.1.

Fig. 2. Rabi oscillations between two X-points, see also movie Rabi.avi. Intensity of the initially Gaussian beam after propagation of t=6 in real (a) and Fourier (b) spaces. The dynamics of the powers P1,2 of two oscillating beams is shown in (c), see text for the details. [Media 1]

Fig. 3. Quasi-one-dimensional tunnelling through the X-point with the tilt directed along b1, see also movie 1D.avi. (a,b) Intensity in the real (top) and Fourier (bottom) spaces are shown for t=0 in (a) and t=10 in (b), corresponding dynamics of beam powers is shown in (d, solid lines). Solutions to the LZM system Eq. (13) are shown in (c) and (d, dashed lines). See text for the details and parameter values. [Media 2]

Fig. 4. (a) The structure of the Bloch bands along the ΓM-direction in a neighborhood of the M-point. The distance between the quasi-degenerate Bloch bands (E2-E1 for -t≫1 and E3-E1 for t≫1) is equal to 3ε1I0/4. (b–d) Symmetric tunnelling through the M-point [24] with the tilt directed along b1+b2, see text and the movie . [Media 3]

Fig. 6. (a) The six resonant Γ-points (indicated by Γj with j=1, …, 6) forming an extended Brillouin zone (solid hexagon). The reciprocal lattice vectors relating the Γ1-point to the other resonant Γ-points are indicated. (b–d) solutions to the LZM systems: (b) corresponds to Eq. (32), (c) to system 1 of (35) and (d) to system 2 of (35). The dashed lines show the analytical results; c1(0)=1 in all cases.

Fig. 8. Three-fold symmetric Zener tunnelling of a vortex beam, see the movie V-1.avi. (a) Intensity and (b) phase of the initial beam with the topological charge m=-1 in the vicinity of the M-point, see (c). Intensities in the Fourier domain (c,d) are identical for both, vortex, m=+1, and antivortex, m=-1, while they differ significantly in the real space, compare (e,g) and corresponding phases (f,h). [Media 6]